Department of Mathematics, Shanghai University, Shanghai 200444, People's Republic of China.
Chaos. 2010 Mar;20(1):013127. doi: 10.1063/1.3314277.
In recent years, fractional(-order) differential equations have attracted increasing interests due to their applications in modeling anomalous diffusion, time dependent materials and processes with long range dependence, allometric scaling laws, and complex networks. Although an autonomous system cannot define a dynamical system in the sense of semigroup because of the memory property determined by the fractional derivative, we can still use the Lyapunov exponents to discuss its dynamical evolution. In this paper, we first define the Lyapunov exponents for fractional differential systems then estimate the bound of the corresponding Lyapunov exponents. For linear fractional differential system, the bounds of its Lyapunov exponents are conveniently derived which can be regarded as an example for the theoretical results established in this paper. Numerical example is also included which supports the theoretical analysis.
近年来，由于分数阶微分方程在模拟反常扩散、时变材料和具有长程相关性、异速生长律和复杂网络的过程中的应用，它们引起了越来越多的关注。尽管由于分数导数决定的记忆特性，自治系统不能按照半群的意义定义动力系统，但我们仍然可以使用李雅普诺夫指数来讨论其动力演化。在本文中，我们首先定义分数阶微分系统的李雅普诺夫指数，然后估计相应的李雅普诺夫指数的界。对于线性分数阶微分系统，方便地导出了其李雅普诺夫指数的界，这可以看作是本文所建立的理论结果的一个例子。还包括数值例子，支持理论分析。
